Epistemology is the study of knowledge and those things closely related to it: justification, what it takes for you to be justified, the relation between knowledge and justification, whether you can have any justified beliefs at all, and if so, how you come to know (or justifiably believe) things, how you can use what you know (or justifiably believe) to come to know (or justifiably believe) other things, and the value of knowledge.
The literature on epistemology is vast. Here's a very brief summary of some epistemological discussions. Concerning knowledge, many epistemologists think knowledge is justified true belief, where the justification you have is linked to the truth of the matter in the right kind of way, though what this way is a matter of debate; some epistemologists think knowledge can't be analyzed this way. Concerning justification's relation to knowledge, some epistemologists think we don't need to be justified to know, and some think we do need to be. Concerning what it takes to be justified, some epistemologists think that what it takes for you to be justified are only factors internal to the believer (Internalists). Others think it takes an external factor, like reliable or well-functioning cognitive faculties (Externalists). Concerning justification, epistemologists differ on whether in any case you there is only one doxastic attitude you are justified in adopting toward a proposition, or whether in some cases more than one doxastic attitude is permissible (as Permissivists maintain). Closely related to the question of “permissivism,” is the question of peer disagreement. Some epistemologists argue that in the face of peer disagreement, one is justified in sticking to one’s guns (as the Steadfast View maintains); others argue that one ought to be conciliatory, moving in the direction of the peer with whom one disagrees (as Conciliationists maintain). Skeptics argue that we can't have any justified beliefs at all, and many epistemologists reply to the skeptic's arguments. Concerning how we use what we know (justifiably believe) to come to know (justifiably believe) other things, some epistemologists (Foundationalists) argue that there are bedrock propositions that we know (justifiably believe), and we build our knowledge (justified beliefs) on those. Foundationalists often take perception, introspection and rational intuition as sources of foundational knowledge, in contrast to testimony (the role of memory being a matter of persistent dispute). Other epistemologists (Coherentists) argue that there aren't bedrock propositions; rather, a set of beliefs is justified as a whole, and several beliefs can be mutually supporting. Concerning the value of knowledge, some argue that knowledge is intrinsically valuable. Others have argued that knowledge is valuable because of the role it plays in practical reasoning, and others argue that knowledge isn't more valuable than justified and true belief, but there are other epistemic states such as understanding, that do have value above their proper subparts. If knowledge is valuable, then treating genuine knowers as if they did not know or depriving people of knowledge, for instance because they belong to an outgroup, can be a form of epistemic injustice.